ANALYTICAL STUDY OF THE PERFORMANCE SURFACE OF BLIND EQUALIZER IN A COSINE MODULATED MULTICARRIER COMMUNICATIONSYSTEM Lekun Lin and Behrouz Farhang-Boroujeny Department of Electrical and Computer Engineering University of Utah Salt Lake City, UT84112 e-mails: llin@eng.utah.edu and farhang@ece.utah.edu. ABSTRACT A recent development in the literature has proposed a cosine modulated filter bank-based multicarrier modulation tech- nique with blind equalization capability. However, conver- gence studies of the proposed blind equalizer has been car- ried out through computer simulations only. In this paper, we present a thorough study of the blind equalizer by ana- lyzing its associated cost function and show that it has two global minimum and two saddle points. 1. INTRODUCTION Multicarrier modulation (MCM) has attracted considerable attention in recent years as a practical and viable technol- ogy for high-speed data transmission over spectrally shaped noisy channels [1, 3]. The discrete multitone/orthogonal frequency division multiplexing (DMT/OFDM) has been rec- ognized as the most cost effective realization of multicarrier transceivers in both wired [2], and wireless [3] channels. Cosine modulated filter banks (CMFB) working at max- imally decimated rate, on the other hand, are well under- stood and widely used for signal compression [5]. More- over, the use of CMFB to multicarrier data transmission over digital subscriber lines (DSL) has been widely addressed in the literature, under the common terminology of discrete wavelet multitone (DWMT) [4]. The major problem with DWMT is the need for a set of special equalizers, one per subchannel. These equalizers that are referred to as linear combiners [4] are two dimen- sional equalizers that span across time and frequency. Each linear combiner usually needs at least 21 taps (7 taps along time and 3 taps along frequency axis) to perform satisfac- torily. This relatively large number of coefficients per lin- ear combiner has the disadvantages of high computational complexity and slow convergence. These difficulties have made DWMT non-attractive to industry, even though it of- fers higher bandwidth efficiency (because of absence of cyclic extensions) and more immunity to narrowband interference. A revisit of DWMT has been made recently [7, 8]. This new study has shown that a modification to the receiver structure in DWMT allows deployment of equalizers that require only two taps per subchannel. Moreover, a blind algorithm that can be used for adaptation of such equal- izers has been proposed. Extensive computer simulations presented in [7, 8] show that the proposed blind equalizer converges to one of its global minima if initialized prop- erly. In another study [9], we derived analytical expression for the performance function of the blind equalizer, found it has four critical points, from which two are global min- ima. However, the nature of the other critical points could be only studied graphically. In this paper, we complete our study by proving that the latter are saddle points. 2. SYSTEM MODEL The multicarrier modulation system that has been proposed in [7, 8] is based on the assumption that the number of sub- channels is sufficiently large so that each subchannel can be approximated by a flat gain. With this assumption, each subchannel of the system may be modeled as in Fig. 1. The input to the subchannel is a complex variable whose real part is the transmitted PAM signal, , and its imaginary part, , arises from intersymbol interference (ISI) from the same subchannel and interchannel interference (ICI) from the adjacent subchannels. Since is a combination of a large number of random variables (ISI and ICI compo- nents), it is shown in [7, 8] that it can be approximated by a Gaussian random variable with the same variance as but independent of . The channel is modeled by the complex gain and the channel noise is . We assume that and are independent Gaussian noise with variance . Re denotes taking the real-part of. 3. BLIND EQUALIZATION Exploring Fig. 1 reveals that ignoring the noise term, the equalizer role is to adjust the phase of the received signal IV - 337 0-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005 ➠ ➡